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**Noncommutative localization in topology.**
*(English)*
Zbl 1125.55004

Ranicki, Andrew (ed.), Noncommutative localization in algebra and topology. Proceedings of the workshop, Edinburgh, UK, April 29–30, 2002. Cambridge: Cambridge University Press (ISBN 0-521-68160-X/pbk). London Mathematical Society Lecture Note Series 330, 81-102 (2006).

In this beautiful paper the author gives a panorama on some very interesting applications in topology of P. M. Cohn’s noncommutative localization concept, recently introduced in noncommutative algebra (see works quoted in the references). In fact, the classical point of view of localization must be considerably revised in order to be used in algebraic topology applications.

Let \(A\) be a (not necessarily commutative) ring and let \(\Sigma\equiv\{s:P\to Q\}\) be a set of morphisms of finitely generated projective \(A\)-modules. A ring morphism \(\phi:A\to R\) is \(\Sigma\)-inverting if \(1\otimes s:R\otimes_{A}P\to R\otimes_{A}Q\) is an isomorphism of finitely generated projective \(R\)-modules, \(\forall s\in\Sigma\). The category of \(\Sigma\)-inverting ring homomorphisms \(A\to R\) has an initial object, denoted \(A\to\Sigma^{-1}A\), i.e., any \(\Sigma\)-inverting ring homomorphism \(A\to R\) factors uniquely as \(A\to\Sigma^{-1}A\to R\).

The ring \(\Sigma^{-1}A\) is called noncommutative localization or universal localization of \(A\) inverting \(\Sigma\). (In the special case, where \(A\) is a commutative ring and \(S\subset A\) is a multiplicatively closed subset, one obtains the classical localization on \(S\), by taking \(P=Q=A\) and the morphisms \(s:A\to A\) to be inverted are given by right multiplication by \(s\in S\). Thus \(\Sigma\cong S\) and \(\Sigma^{-1}A\cong S^{-1}A\). This works also if \(A\) is not commutative but \(S\subset Z(A)\) is contained in the centre \(Z(A)\) of \(A\). In such a case \(\Sigma^{-1}\) is the commutative localization of \(A\).)

The applications to topology, considered in this paper, involve homology with coefficients in a noncommutative localization. The paper, after a short introduction, splits into two more sections. 1. This gives a survey of some applications of noncommutative localization to topology: finitely dominated spaces, codimension \(1\) and \(2\) embeddings, homology surgery theory, open book decompositions and circle-valued Morse theory. 2. This reports on some of the author’s applications of noncommutative localization to chain complexes over generalized free products. Relations with works by Bergman, Schofield, Waldhausen and Cappell are given, in connection to a noncommutative localization interpretation of the Seifert-van Kampen and Mayer-Vietoris theorems.

For the entire collection see [Zbl 1108.13001].

Let \(A\) be a (not necessarily commutative) ring and let \(\Sigma\equiv\{s:P\to Q\}\) be a set of morphisms of finitely generated projective \(A\)-modules. A ring morphism \(\phi:A\to R\) is \(\Sigma\)-inverting if \(1\otimes s:R\otimes_{A}P\to R\otimes_{A}Q\) is an isomorphism of finitely generated projective \(R\)-modules, \(\forall s\in\Sigma\). The category of \(\Sigma\)-inverting ring homomorphisms \(A\to R\) has an initial object, denoted \(A\to\Sigma^{-1}A\), i.e., any \(\Sigma\)-inverting ring homomorphism \(A\to R\) factors uniquely as \(A\to\Sigma^{-1}A\to R\).

The ring \(\Sigma^{-1}A\) is called noncommutative localization or universal localization of \(A\) inverting \(\Sigma\). (In the special case, where \(A\) is a commutative ring and \(S\subset A\) is a multiplicatively closed subset, one obtains the classical localization on \(S\), by taking \(P=Q=A\) and the morphisms \(s:A\to A\) to be inverted are given by right multiplication by \(s\in S\). Thus \(\Sigma\cong S\) and \(\Sigma^{-1}A\cong S^{-1}A\). This works also if \(A\) is not commutative but \(S\subset Z(A)\) is contained in the centre \(Z(A)\) of \(A\). In such a case \(\Sigma^{-1}\) is the commutative localization of \(A\).)

The applications to topology, considered in this paper, involve homology with coefficients in a noncommutative localization. The paper, after a short introduction, splits into two more sections. 1. This gives a survey of some applications of noncommutative localization to topology: finitely dominated spaces, codimension \(1\) and \(2\) embeddings, homology surgery theory, open book decompositions and circle-valued Morse theory. 2. This reports on some of the author’s applications of noncommutative localization to chain complexes over generalized free products. Relations with works by Bergman, Schofield, Waldhausen and Cappell are given, in connection to a noncommutative localization interpretation of the Seifert-van Kampen and Mayer-Vietoris theorems.

For the entire collection see [Zbl 1108.13001].

Reviewer: Agostino PrĂˇstaro (Roma)